Matrix multiplication and tensor contraction cost

The computational cost of a matrix multiplication \(A\cdot{B}\) reads \(\mathfrak{C}\left(A\cdot{B}\right)=d_{0}d_{1}d_{2}\) with \(A\) is a \(d_{0}\times{d}_{1}\) matrix and \(B\) is a \(d_{1}\times{d}_{2}\) matrix …

A generic method to construct matrix product operator

Written by Wayne Zheng in PHYSICS on Thu 16 September 2021. Tags: tensor network,

Preliminary

From a many-body operator (MBO) to a matrix product operator (MPO)

We are talking about quantum many-body models on a lattice \(\Lambda\). A lattice \(\Lambda\) is a graph consisting of \(|\Lambda|\) sites, which are always labeled in a one-dimensional array \(\left\{0, \dots, |\Lambda|-1\right\}\) in a specific …

How to wisely contract a tensor network

Written by Wayne Zheng in PHYSICS on Thu 21 May 2020. Tags: tensor network,

Introduction

Tensors and their contraction

Tensor is a higher-dimensional generalization of the matrix, which has two indices such as \(M_{ij}\). For example, a four-dimensional tensor can be obtained by the tensor product of two matrices \(\left(A\otimes B\right)\_{ij}=C_{ij}=C_{k\cdot d_{l}+l …